Bayesian Personalised Ranking Loss

Compute the Bayesian personalised ranking (BPR) loss for implicit feedback.

Many systems only see clicks or views, never explicit ratings. BPR does not try to predict a rating. It learns to rank: for a user u, an item p they interacted with should score higher than an item n they did not. The score of an item is the dot product of the user and item vectors. The loss for one triple is

x     = u . p - u . n
loss  = -log(sigmoid(x)) = log(1 + exp(-x))

Return the mean loss over the batch of triples. Compute log(1 + exp(-x)) in a stable way, for example with np.logaddexp(0, -x), so that large negative x does not overflow.

Function signature

def bpr_loss(user_vecs, pos_vecs, neg_vecs):

Example

Input

bpr_loss(np.array([[1.0, 0.0]]), np.array([[1.0, 0.0]]), np.array([[0.0, 1.0]]))

Output

0.31326169

Reasoning

The positive item scores 1 and the negative item scores 0, so x = 1. log(1 + exp(-1)) = 0.3133. When the positive item wins by a wide margin the loss goes to 0. When the two items tie, the loss is log 2.

Constraints and notes

• All three inputs have shape (batch, k). Row b of each forms one triple.
• Return a plain float.